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Find equation of motion using. Lagrangian given equation is $L'=\frac{m}{2}(a\dot{x}^2+2b\dot{x}{y}+c\dot{y}^2)-\frac{k}{2}(ax^2+2bxy+cy^2)$

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$$L'=\frac{m}{2}(a\dot{x}^2+2b\dot{x}{y}+c\dot{y}^2)-\frac{k}{2}(ax^2+2bxy+cy^2)$$
where a, b, and c are arbitrary constants but subject to the condition that $b^2 − ac \ne 0$. What are the equations of motion? Examine particularly the two cases $a = 0 = c$ and $b = 0$, $c = −a$. What is the physical system described by the above Lagrangian? Show that the usual Lagrangian for this system as defined by Eq. (1.56) is related to $L'$ by a point transformation (cf. Derivation 10). What is the significance of the condition on the value of $b^2 − ac$?
image of book
$L=T-U \ \text{Eq.(1.56)}$

It's two dimensional. If there was only $x$ than, it was easier to differentiate. But, I was thinking what I should do now. Then, all the things I had done that is differentiating respect to $x$ and $y$ both.

$$\frac{d^2 L}{dx dy}=-kax-bkx-bky-cky$$ $$\frac{d^2 L}{d\dot{x}d\dot{y}}=am\dot{x}+bm\dot{y}+bm\dot{x}+cm\dot{y}$$ $$\frac{d}{dt}(\frac{d^2 L}{d\dot{x}d\dot{y}})=am\ddot{x}+bm\ddot{y}+bm\ddot{x}+cm\ddot{y}$$ $$\frac{d}{dt}(\frac{d^2 L}{d \dot{x} d\dot{y}})-\frac{d^2}{dxdy}=0$$ $$am\ddot{x}+bm\ddot{y}+bm\ddot{x}+cm\ddot{y}=-(kax+bkx+bky+cky)$$ How can I find equation of motion for that? (What are the equations of motion?) Should I find a equation for $x$ at first then another equation for $y$?

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Your mistake is that you did a second derivative of $L$, taking the derivative according to both degrees of freedom together. Instead you need to make a separate equation for each degree of freedom.

Since you have two degrees of freedom ($x,y$), you get two equations:

\begin{align} \frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial\dot x} - \frac{\partial L}{\partial x} &= 0\\ \frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial\dot y} - \frac{\partial L}{\partial y} &= 0 \end{align}

The two equations you get this way are the equations of motion.

Another mistake is that you are writing total derivatives where you would need to write partial ones; the only total derivative is the time derivative. Since you are using them as if they were partial derivatives, this is inconsequential in your calculation, though.

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